Scheme of Studies - Rising Education · MTH-506 Mechanics 4(4-0) MTH-508 Functional Analysis 4(4-0) MTH-510 Differential Geometry 4(4-0) Semester 7 PURE MTHEMATICS Course Code Course

GC UNIVERSITY, FAISALABAD

Scheme of Studies BS Honors in Mathematics

8 Semesters / 4 years Degree Program

for the year 2012 and onwards

Department of Mathematics

Scheme of Studies BS Hons in Mathematics

2 GC University, Faisalabad

Scheme of Studies BS Honors in Mathematics

Semester 1

Course Code Course Title Credit Hours

ISL-302 Islamic Studies 2(2-0)

ENG-321 English-I 3(3-0)

MTH-301 Calculus-I 4(4-0)

MTH-303 Mathematical Method-I 4(4-0)

PHY-325 Physics-I 4(3-1)

Semester 2


PST-322 Pakistan Studies 2(2-0)

ENG-322 English-II 3(3-0)

STT-322 Statistics-I 4(4-0)

MTH-302 Calculus-II 4(4-0)

MTH-304 Mathematical Method-II 4(4-0)

Semester 3


MTH-401 Vector Calculus 2(2-0)

MTH-403 Mechanics I 4(4-0)

MTH-405 Mechanics II 4(4-0)

STT-421 Statistics-II 3(3-0)

PHY-425 Physics-II 3(3-0)

Semester 4


MTH-402 Metric and Topological Spaces 4(4-0)

MTH-404 Differential Equations 4(4-0)

MTH-406 Numerical Analysis 2(2-0)

MTH-408 Operations Research 2(2-0)

CSI-422 C++ 4 (4-0)

Semester 5


MTH-501 Real Analysis I 4(4-0)

MTH-503 Complex Analysis 4(4-0)

MTH-505 Vector and Tensor Analysis 4(4-0)

MTH-507 Algebra-I 4(4-0)

MTH-509 Point Set Topology 4(4-0)



Semester 6


MTH-502 Real Analysis-II 4(4-0)

MTH-504 Algebra-II 4(4-0)

MTH-506 Mechanics 4(4-0)

MTH-508 Functional Analysis 4(4-0)

MTH-510 Differential Geometry 4(4-0)

Semester 7

PURE MTHEMATICS


MTH-601 Advanced Group Theory 4(4-0)

MTH-603 Advanced Set Theory 4(4-0)

Optional Paper (3 out of Following)

MTH-605 Mathematical Statistics-I 4(4-0)

MTH-607 Continuous Groups 4(4-0)

MTH-609 Theory of Modules 4(4-0)

MTH-611 Algebraic Topology 4(4-0)

MTH-613 Advanced Topology 4(4-0)

MTH-615 Numerical Analysis-I 4(4-0)

MTH-617 Linear Algebra 4(4-0)

MTH-619 Rings and Fields 4(4-0)

APPLIED MTHEMATICS


MTH-621 Fluid Mechanics-I 4(4-0)

MTH-623 Advanced Mathematical Methods 4(4-0)


MTH-605 Mathematical Statistics-I 4(4-0)

MTH-625 Special Theory of Relativity 4(4-0)

MTH-627 Operations Research 4(4-0)

MTH-629 Quantum Mechanics 4(4-0)

MTH-631 Soft Ware Engineering 4(4-0)

MTH-615 Numerical Analysis-I 4(4-0)



Semester 8

PURE MTHEMATICS


MTH-602 Measure Theory 4(4-0)

MTH-604 Advanced Functional Analysis 4(4-0)


MTH-606 Rings & Modules 4(4-0)

MTH-608 Theory of Numbers 4(4-0)

MTH-610 Mathematical Statistics-II 4(4-0)

MTH-612 Numerical Analysis-II 4(4-0)

MTH-614 Special Functions 4(4-0)

MTH-616 Theory of Optimization 4(4-0)

MTH-628 Project 4(4-0)

APPLIED MTHEMATICS


MTH-618 Fluid Mechanics-II 4(4-0)

MTH-620 Partial Differential Equations 4(4-0)


MTH-622 Theory of Elasticity 4(4-0)

MTH-624 Electromagnetism 4(4-0)

MTH-610 Mathematical Statistics-II 4(4-0)

MTH-612 Numerical Analysis-II 4(4-0)

MTH-614 Special Functions 4(4-0)

MTH-616 Theory of Optimization 4(4-0)

MTH-628 Project 4(4-0)

MTH-626 C ++ 4(4-0)

The Project of BS (H) Mathematics will be offered as an optional paper to not more than 50

% of the class strength and only to those who obtain at least 65 % marks onthe basis of their

performance in V & VI Semesters.



Course Number Title Credit Hours Marks

ISL-302 Islamic Studies 2(2-0) 40


ENG-321 English-I 3(3-0) 60

ENGLISH-I

MID EXAM

Sentence, Article, Auxiliary Verbs, Moods of verb, Tenses, Parts of speech, Types of

sentences, Sentence Structure (Subject + Predicate), Phrase, Clause, Infinitives, Sequence of

Sentences, Transitive and Intransitive verb.

FINAL EXAM

Conditional Sentences, Comprehensive, Précis, Translation (U to E), Tenses, Letters,

Application (For Jobs)

RECOMMENDED BOOKS

A.J. Thomson and A.V. Martinet, A Practical English Grammar, Oxford University Press

A.J. Thomson and A.V. Martinet, Exercise Book 1, Oxford University Press.


MTH-301 Calculus-I 4(4-0) 80

CALCULUS-I

The Real Number System. Axiom of Addition, Subtraction and Multiplication. Existence of

Irrational Numbers. The Order Properties of R. Absolute Value of a Real Number. The Real

Line Inequalities. Functions Limits of Functions, Continuity and Derivability. Notation and

Geometrical Interpretation of the Derivative, Derivatives of Trigonometric, Inverse

Trigonometric, Logarithmic, Exponential, Hyperbolic, Inverse Hyperbolic and Implicit

Functions, Approximation, Related Rates, Higher Derivates, Leibniz Theorem and its

Application, Rolle’s Theorem, Mean Value Theorem, Maclurin’s and Taylor's Expansion

series with application. Cauchy’s Mean Value Theorem their Geometrical Interpretation.

Increasing and Decreasing Functions. Indeterminate forms. Anti-Derivates, Integration by

Substitution, Integration by Parts. Integration of Rational, Irrational and Trigonometric

Functions. The Definite Integrals and their Properties. Improper Integrals. Reduction

Formulas

RECOMMENDED BOOKS

George B. Thomas, Calculus and Analytic Geometry.

Spiegal M. P; Advance Calculus New York. Mc Graw Hill Book Company

Dr. S. M. Yousaf and Muhammad Amin Ch, Calculus with Analytic Geometry Ilmi Kitab

Khana Urdu Bazar Lahore.




MTH-303 Mathematical Method-I 4(4-0) 80

MATHEMATICAL METHOD-I

Complex Numbers, Addition, Multiplication, Modulus, Conjugate and Argument of a

Complex Number, the Argand’s Diagram, Polar Form of A Complex Number, DeMoivre’s

Theorem and its Application, Exponential, Trigonometric, hyperbolic, Logarithmic, Inverse

Hyperbolic and Inverse Trigonometric Functions. Complex Powers, Simple case of

Summation of Series. Algebra of Matrices. Inverse of a Matrix. Elementary Row and

Column operations. System of Linear Equations. Gaussian and Gauss Jordan Elimination

Methods. Consistency Criterion. Determinant of Square Matrices Properties of Determinants.

Determinate of the Transpose Matrix Sequences, Basic Theorems on sequences, Infinite

series P-Test, the divergence Test, The Basis Comparison Test, The Integral Test, The

Cauchy’s Root test. The D/e Alembert Ratio Test, Alternating series, Absolute and

conditional Convergence. Convergence of Power Series. Differentiation, Integration,

Addition and Multiplication Power Series.

RECOMMENDED BOOKS

Murrsy R.Spegal Complex Variables Mc. Graw-Hill book Company New York.

Aitken A.C Determinant and Matrices Edinburgh: Olever and Boyd.

Spiegal M.R; Advance Calculus Mc Graw-Hill Book Company New York,

S.M. Yousaf and M. Amin, Mathematical Methods, Ilmi Kitab Khana latest edition Urdu

Bazar Lahore.


PHY-325 Physics-I 4(3-1) 80

PHYSICS-I

VECTOR ANALYSIS: Vector Operations, Gradient of a Scalar field, Divergence of a

vector field, Curl of a vector Field, Divergence theorem, Stoke’s theorem.

Suggested Level: Ch # 1 of Mechanics by Tirmizi, 2nd

Edition

SOUND WAVES: Wave motion and Sound with Beats, Doppler Effect and its application

Suggested Level: Ch # 1 of Wave & Oscillation by Dr. Tahir Hussain T.I New Edition

FLUID DYNAMICS: Bernoulli’s Equations, Applications, of Bernoulli’s Equation,

Suggested Level: Ch # 15 of Physics by R.H.K, 4th

Edition

WORK AND ENERGY: Introduction, Work-Energy Theorem,


Edition

ELECTROMAGNETIC SPECTRUM: Radio waves, Microwaves (terrestrial

Microwaves & Satellite Microwave)


Edition

LASER: Principal of Laser, Characteristics and use of laser


Edition



FIBER OPTICS: Principals and working of fiber optic, Advantages and Disadvantages,

Computer Networks and Fiber Optic


Edition


PST-322 Pakistan Studies 2(2-0) 40

PAKISTAN STUDIES

The main objective of this paper is to familiarize the students with the freedom movement,

Pakistan history and the foreign policy of Pakistan

Aligrah movement with special reference to its Educational and Political Activities,

Formation of Muslim League and its role in the creation of Pakistan, Khilafat Movement and

its impact upon the political culture of the sub-continent, Comparative study of Nehru Report

and 14 points of Jinnah, Iqbal’s Allahabad Address, and its Significance, A brief overview of

constitutional development in Pakistan, Pakistan Resolution and the creation of Pakistan,

Quaid-i-Azam, as a political leader and founder of Pakistan, Pakistan’s geo-strategic position

and its importance, Constitutional development in Pakistan: A brief analysis, Pakistan’s

political culture and problems of democracy, Human rights in Pakistan, Major determinants

of the foreign policy of Pakistan

RECOMMENDED BOOKS

M Ikram Rabbani, Pakistan Studies

M Ikram Rabbani, Current Affairs

Ahmed Saeed, Track to Pakistan

Safdar Mahmood, Political Roots and Development

Waheed-uz-Zaman, Towards Pakistan

Dr. M Imtiaz Shahid, An advance study in Pakistan Affairs.




ENG-322 English-II 3(3-0) 60


STAT-322 Statistics-I 4(4-0) 80

STATISTICS

Meanings of Statistics, Main branches of Statistics (Theoretical and Applied), Meanings of

Descriptive and Inferential Statistics, Population and Sample, Types of Variables,

Measurement Scales. Sources of statistical data in Pakistan. Description of data by frequency

Tables and Graphs. Stem and Leaf display and Box Plots.

Measures of central tendency: Arithmetic Mean, Geometric Mean, Harmonic Mean, Mode,

Median and Quantiles. Properties of Arithmetic mean with proofs. Weighted arithmetic

mean. Empirical Relation between Mean, Median and Mode. Relative Merits and Demerits

of Various Averages.

Measures of Dispersion. Absolute and Relative Measures, Range, Semi-inter Quartile Range,

Mean Deviation, Variance, Standard Deviation. Coefficient of Variation. Coefficient of

Mean Deviation, Coefficient of Quartile Deviations, Properties of Variance and Standard

Deviation with Proofs. Standardized Variables, Moments, Moment Ratios, Sheppard’s

Correction, Kurtosis and Skewness.

RECOMMENDED BOOKS

Wonnacott, T.H and Wannacott, R.J (1990). Introductory Statistics. Jhon Wily & Sons. New

York.

Walpole, R.E (2001). Introduction to Statistics. Macmillan Publishing Company.

Rauf, M (2001). Polymers Modern Statistics. Polymer Publication, Urdu Bazar, Lahore.

Chaudhray, S.M and Kamal, S. (2002). Introduction to Statistical Theory. Ilmi Kitab Khana,

Urdu Bazar, Lahore.

Mood, A.M. Gray-bill, F.A. Boes, D.C. Introduction to the theory of statistics (2nd

edition)

1986, McGraw-Hill book company New York.




MTH-302 Calculus-II 4(4-0) 80

CALCULUS-II

General Conics, Tangents and Normal, Polar Coordinates Conic in Polar Coordinates,

Relationship between Rectangular and polar system. Simple curve tracing, in polar and

Cartesian coordinates. Tangents and Normal in Cartesian and polar Coordinates, Parametric

representation of Curve. Asymptotes (Rectangular and polar Curves) Maxima, Minima,

Singular points (Node, Cups, Isolated pts), Quadrature, Lengths of Arcs, Intrinsic Equations,

Curvature, Circle of Curvature, Envelopes and Evolutes, Straight Line in R3 Angle between

two Straight lines, Distance of a pt. form a line, Equations for planes. Angle between two

planes Straight Lines and Planes. Shortest Distance between two Straight Lines. Functions of

several Variables partial Derivatives. Homogeneous Functions, Differentials, tangent Plane

and the Normal Line. Exterma of Functions of two Variables. Double Integrals, Triple

integrals Area and volume by double integrals, Volume and Area of Surface of Revaluation.

RECOMMENDED BOOKS

George B. Thomas, Calculus And Analytic Geometry

Spiegal M.R; Advance Calculus Mc Graw Hill Book Company New York.

S.M. Yusuf and M. Amin, Calculus with Analytic Geometry, Ilmi Kitab Khana Urdu Bazar

Lahore.


MTH-304 Mathematical Method-II 4(4-0) 80

MATHEMATICAL METHOD-II

Groupoid, Semi Group, Groups, Properties of Groups, order of Group, Order of an element,

Subgroups, Cyclic Groups, Cyclic Subgroup, Cossets and Lagrange’s Theorem.

Permutations, Cycles, Transpositions, Order of a permutation, Introduction of Rings and

Fields. Definition, Example and Elementary Properties of a vector space, Subspaces, Linear

Combination, Linear Dependence, Basis of Vector spaces. Linear Transformation. Matrix of

Linear Transformation. Inner Product Spaces (Definition and Example) Orthogonality,

Orthogonal Matrices, Eigen Values and Eigen Vectors, Similar Matrices, Digitalization of

Matrices.

RECOMMENDED BOOKS

Murrsy R.Spegal Complex Variables Mc. Graw-Hill book Company New York.

Aitken A.C Determinant and Matrices Edinburgh: Olever and Boyd.

Spiegal M.R; Advance Calculus Mc Graw-Hill Book Company New York,

S.M. Yousaf and M. Amin, Mathematical Methods, Ilmi Kitab Khana latest edition.




MTH-401 Vector Calculus 2(2-0) 40

VECTOR CALCULUS

Scalars and vectors. Representation of Vectors. Types of vectors. Addition & Subtraction of

Vectors. Properties of vector addition. Multiplication of a Vector by a Scalar. The unit

Vectors i, j, and k.

Vector or Cross Product of two Vectors. Vectors Area of a Triangle. Product of three

Vectors. Geometrical Interpretation of Scalar Triple Product, Condition for four points to be

Coplanar. Vector Triple Product, Scalar and vector Products of Four Vectors.

Scalar point function, Vector point function, Continuous Function, Differentiation of Vector

Function. Geometrical Meaning of dr/dt. Derivative of a vector function in terms of its

components. Integration of Vector Functions. Partial Derivatives and Differentials.

Differential Operator Del , Gradient, Divergence and Curl of a Vector Point Function,

Vector Perpendicular to Level Surface. Directional Derivatives.

RECOMMENDED BOOKS

Smith G.D. Vector Analysis Oxford University Press, Oxford.

Spiegel M. R, Vector Analysis 1974, McGraw Hill New York.

Dr. Munawar Hussain Elementary Vector Analysis, 5th

Edition 2006, Caravan Book House

Lahore.

Dr. Nawazish Ali Shah, Vector and Tensor Analysis, A-One Publishers Lahore.

Dr. S. M. Yousaf Vector Analysis Ilmi Kitab Khana Urdu Bazar Lahore.


MTH-403 Mechanics-I 4(4-0) 80

MECHANICS-I

Components of Force. Composition of Concurrent forces, the ),( Theorem

Equilibrium of a Particle. Moment of a force about a point. Virginan’s Theorem. Couples.

Equivalent Couples. Composition of Couples Resolution of a Force into a force and a couple.

Resolution of a system of Force into a force and a couple. Resolution of a system of Parallel

Force Condition of Equilibrium of Coplanar force system. Types of forces. Direction of

Forces of Constraints. Equilibrium of 3 coplanar forces. Laws of Friction, Cone of Friction.

Role of Friction. Equilibrium of a Particle on a rough inclined plane. Equilibrium of a

particle on a rough Horizontal Plane. Definition of Virtual Work Applied Forces and Forces

of Constraints. Virtual Displacement and virtual Work. Workless constraints Principle of

Virtual Work, for Single particle, set of Particles and for a Rigid Body.

BOOKS RECOMMENDED Synge & Griffith, Principles of Mechanics, McGraw Hill Book Company Inc., New York.

D.T. Greenwood, Principles of Dynamics, Prentice Hall, Inc.

W. Huser, Introduction to Principles of Mechanics, Addison Wesley, New York.

R.A Becker, Introduction to Theoretical Mechanics, McGraw Hill Book Company, Inc., New York.



F. Chorlton, A Text Book of Dynamics.

K .L. Mir, Theoretical Mechanics Ilmi Kitab Khana.


MTH-405 Mechanics-II 4(4-0) 80

MECHANICS-II

Velocity and Acceleration. Cartesian components of velocity and acceleration. Tangential

and Normal Components of Velocity and Acceleration. Transverse and Radial Components

of Velocity and Acceleration. Motion with constant Acceleration. Motion with Variable

Acceleration. Graphical Methods. Motion of a Free Particle along the vertical Line, Simple

Harmonic Motion. The Nature of Simple Harmonic Motion. Geometrical Representation.

Speed of the Projectile Parabola of Safety. Range on the Inclined Plane. Maximum range on

horizontal and Inclined Plane. Time Period, Maximum Height. Motion under a Central Force.

Elliptic Orbit Under a central force. Polar form of the orbit. Apse and apsidal distance.

Planetary orbits. Kepler’s Laws. Damped Harmonic Oscillator, Damped force oscillations,

Vertical motion with air Resistance

RECOMMENDED BOOKS

Synge & Griffith, Principles of Mechanics, McGraw Hill Book Company Inc., New

York.



R.A Becker, Introduction to Theoretical Mechanics, McGraw Hill Book Company,

Inc., New York.


K .L. Mir, Theoretical Mechanics Ilmi Kitab Khan


STT-421 Statistics-II 3(3-0) 60

STATISTICS-II

Introduction of Simple Linear Regression. Scatter Diagram. Simple Linear Regression

Model. Assumptions of Linear regression model. Least Squares Estimators. Properties of the

Least Square Regression Line. Standard Errors of Estimates.

Correlation, Pearson’s Product Moment Correlation Coefficient with its application in daily

life. Properties of Correlation Coefficient, Coefficient of Determination, Correlation

Coefficient for group data. Rank Correlation and its properties. Spearman’s Rank Correlation

Coefficient and its derivation. Tied ranks.

Multiple Linear Regression with two Regressors, Coefficient of Multiple Determination,

Standard Error of Estimate. Coefficient of Partial and Multiple Correlations. Relation

between Partial and Multiple Correlation Coefficients.



Curve Fitting by Method of Least Squares. Fitting a straight line, fitting of a Second and

third degree Parabola, Change of Origin and Unit, Exponential Curves, Criteria for a Suitable

Curve, Finding Plausible Values by LS Method.

BOOKS RECOMMENDED

Wonnacott, T.H and Wannacott, R.J (1990). Introductory Statistics. Jhon Wily & Sons. New

York.

Walpole, R.E (2001). Introduction to Statistics. Macmillan Publishing Company.

Rauf, M (2001). Polymers Modern Statistics. Polymer Publication, Urdu Bazar, Lahore.

Chaudhray, S.M and Kamal, S. (2002). Introduction to Statistical Theory. Ilmi Kitab Khana,

Urdu Bazar, Lahore.


PHY-425 Physics-II 3(3-0) 40

PHYSICS-II

Coulomb’s law. Electric field with different charges. Electric flux. Gauss’s law and its

applications. Magnetic field. Magnetic flux. Faraday’s law and its applications. Biot-Savart

law and its applications. Ampere’s circuit law and its applications. Electromagnetic

inductions. Motional electromotive force. Self-inductance with solenoid and toroid. Growth

and Decay of current, in an (R-l and R-C) series circuit. Electromagnetic oscillators. (A-C

and D-C) voltage applied to “inductors and capacitors) Phaser concept. A sinusoidal voltage

applied to an (L-R, R-C, R-l-C) series circuits. Frequency response of (R-L-C) series circuits.

Semi conductors with (P-Type and N-Type) material. Pn-Junction. Resistance. Capacitance.

Inductance. Diodes. Rectifiers. Multi-vibrators. Logic-gates. Generators. Motors.

Transformers.


MTH-402 Metric and Topological Spaces 4(4-0) 80

METRIC AND TOPOLOGICAL SPACES

Distance in R2 metric, definition of metric and metric space, examples, balls, diameters, open

& closed ball, open set & close set. interior points and interior of a set, exterior points and

exterior of a set, closure of a subset, limit points, neighborhood points, boundary points,

sequences and their convergence complete space,

Basic notions of set theory, set operations, extended set operations and indexed families of

sets. relations, equivalence relations, partition, ordering relations, function as relations,

topological spaces; subspaces and relative topology, open sets, closed sets, neighborhood,

interior, exterior boundary and limit points, base and sub base.

RECOMMENDED BOOKS

G.F. Simon, Introduction to Topology and Modern Analysis, 1963, McGraw Hill Book

Company, New York.

J. Willard, General Topology, Addison-Wesley Publishing Company, London.

E. Kreyszig, Introduction to Functional Analysis with Applications, 1978, John Wiley and

Sons.



W. Rudin, Functional Analysis, McGraw Hill Book Company, New York.

N. Dunford and J. Schwartz, Linear Operators (Part-I General Theory), Interscience

Publishers, New York.


MTH-404 Differential Equations 4(4-0) 80

DIFFERENTIAL EQUATIONS

Differential Equations and their classification, Formation of Differential Equations, Initial

values and boundary values Problem, Separation of Variables. Homogenous Differential

Equations, Exact Differential Equations, Differential equations reduceable to homogenous

form, Linear Differential Equations of 1st Order, Bernonlli’s Equations. Orthogonal

Trajectories, Homogenous linear equations, Non-Homogenous Differential Equations with

constant Coefficient, D & Inverse D-1

, Operators, General & Particular Integrals. Cauchy-

Eluer’s equations, Reduction of order, Method of Variation of Parameter’s, Exact Linear

Equations, System of Linear Differential Equations, Power Series Solutions of first order

Differential Equations. Laplace and Inverse Transformations with simple Application to

Differential Equation.

RECOMMENDED BOOKS

Zill D G, Cullen M.R. Differential Equations with Boundary-Value Problems (3rd

Edition),

1997, PWS Publishing Co.

Eisgolts L, Differential Equations and the Calculus of Variations, 1970, Mir Publishers

Moscow.

Jerri A.J Introduction to Integral Equations with Applications, 1985, Marcel Dibber New

York.

Muhammad Amin, Mathematical Methods, 2007, Ilmi Kitab Khana Lahore.


MTH-406 Numerical Analysis 2(2-0) 40

NUMERICAL ANALYSIS

Introduction, Computation error analysis, Study of various iterative methods to solve non-

linear equations with analysis of error, convergence and stability of bisection, false position,

secant, Newton-Raphson and fixed point methods, acceleration of convergence by Aitken

method, solution of system of linear equation by LU decomposition method, cases of failure,

iterative methods, (Jacobi, Gauss Seidel, SOR, SUR) and their convergence analysis, ill

conditioned systems and condition number, interpolation: Review of simple interpolation for

equally spaced data, interpolation by Gauss forward/backward methods, Bessel and Stirling

method with error analysis, Lagrange Interpolation and Newton divided differences formula

with error analysis.

RECOMMENDED BOOKS

Robert-W. Horn Numerical Methods.

Alestair Wood Introduction to Numerical Analysis.



M. Iqbal Numerical Analysis Nation Book of Foundation

S.A. Bhatti, N. A Bhatti, Numerical Methods.

Ch. Muhammad Saleem Numerical Analysis.


MTH-408 Operations Research 2(2-0) 40

OPERATIONAL RESEARCH

Introduction, Brief History, Optimization Techniques, System models and Modeling,

Classification of OR Models, Phases of an OR study. Hungarian Method (Assignment

Problem). Transportation Problem. Classification of Mathematical Programming. Linear

Programming (LP), Formulation of LP problems. Simplex Method, The M-Method and the

Two phase Technique for starting Optimization. Duality and Sensitivity Analysis, Critical

Path Method (CPM), Program Evaluation and review Technique (PERT).

RECOMMENDED BOOK

Hamdy A Taha An Introduction to Operations Research Macmillan Publishing Company

Inc, New York, 1987.

S Kalavathy Operations Research Vikas Publishing House (P) Ltd.

F. S Hiller and G. J Liebraman, Operational Research CBS Publisher and distributors, New

Delhi, 1974.

C. M Harvey, Operation Research, North Holland, New Delhi

Prof. Sr. Saeed Akhtar Bhatti Operations Research: An Introduction

Krajewsky and Ritzman Operations Management Strategy and Analysis.


CSI-422 C++ 4(4-0) 80

C++

History of Computer Languages, Building Blocks of C++. Control Structures. Functions,

Arrays, Pointers, Strings, Object Oriented Concepts e.g. Operator Over landing, Inheritance,

Overriding, Polymorphism & Virtual Functions. Introduction to Java.

RECOMMENDED BOOKS

Herbet Shildt, Turbo C/C++

Eric P. Bloom, The Turbo C++

Robert Lafore, Turbo C

Deitel & Deitel C++ How to Programme




MTH-501 Real Analysis -I 4(4-0) 80

REAL ANALYSIS-I

Algebraic and ordered properties of Real Numbers, Absolute values, Inequalities (Cauchy’s,

Minkoski’s, Bernoulli’s) Properties and concepts of supremum and infimum, Ordered sets,

Fields, Field of Real, The extended real number system, Euclidean spaces, Sequences,

Subsequences, Cauchy sequence, Series of Numbers and their convergence. The

Comparison, Root, Ratio and Integral tests. Absolute and Conditional convergence of infinite

series. Limits and Continuity. Properties of continuous functions. Types of discontinuities.

Differentiable functions. Mean-value theorems, Continuity of derivatives. Partial Derivatives

and Differentiability. Derivative and differentials of Composite functions. The Directional

Derivative, the Laplacian in polar cylindrical and Spherical coordinates.

BOOKS RECOMMENDED Bartle R G and Sherbert DR, Introduction to Real Analysis, 1999 John Wiley New York.

Rudin W. Principles of Mathematics Analysis, 1986 McGraw-Hill New York.

Brabence RL Introduction to Real Analysis, 1997 PWS Publishing Company.

Gaughan ED. Introduction to Analysis (5th

Edition) 1997 Brooks/Cole

Kaplan W. Advance Calculus1984 Addison-Wesley publishing Company.


MTH-503 Complex Analysis 4(4-0) 80

COMPLEX ANALYSIS

Analytic functions, Cauchy-Riemann equations. Power series, Radius of convergence.

Cauchy’s theorem. Cauchy’s integral formula and related theorems. Contour integration.

Singularities, Branch points. Taylor’s and Laurent’s series. Analytic continuation. Residues,

Residue theorem. Fundamental theorem of Algebra. Application of calculus of residues to

infinite products. Conformal transformations and their applications.

BOOKS RECOMMENDED

L. Pennisi, L. Gordon, and S. Lasher, Elements of Complex Variables, Holt Rinehart and

Winston.

R.Churchill, Complex Variables and Application (1996). McGraw Hill.

R.A.Silverman, Complex Analysis with Applications, Prentice Hall, Englewood Cliffs,

N.Jersey.

J. Paliouras, Complex Variables for Scientists and Engineers McMillan

H.R.Chillingworth, Complex Variables, Pergamon Press, Oxford.

L.V. Ahlfors, Complex Analysis McGraw Hill.

M. Iqbal, Complex Analysis (1996). Ilmi Kitab Khana.

K. Kodaira, Introduction to Complex Analysis, Cambridge.

S. Lang, Complex Analysis. 4th

Edition, 2001, Springer.




MTH-505 Vector& Tensor Analysis 4(4-0) 80

VECTOR AND TENSOR

VECTOR ANALYSIS

Curvilinear Coordinates, Scale Factors, Arc length, Area and volume in curvilinear

coordinates. Spherical and Cylindrical coordinates, Expansion formulas of

Gradient, Divergence and Curl of point in curvilinear coordinate, Relation between

orthogonal bases, Curvilinear Coordinates, Spherical and cylindrical coordinates and

their applications, Line, Surface and volume integral. Gausses, Green’s and Stokes

theorem with their application.

CARTESIAN TENSORS

Cartesian Tensor, Law of transformation, from one coordinates system to another

coordinates system. Algebra of Tensor, Sum, Difference, Quotient, Inner product,

Contraction, Contraction theorem with their application, Symmetric Tensor and Skew

Symmetric Tensor, Kronecker Tensor and Levi Civita Tensor, Relation between these

Tensors, Isotropic Tensor, First and Second order Differential Operators in Tensor.

Application of Tensors in Vector Analysis, Proof of Expansions formulae using

tensor formulism.

BOOKS RECOMMENDED

H. Jeffrey, Cartesian Tensors, Cambridge University Press.

F. Charlton, Vector and Tensor Methods, Ellis Harwood. Publisher, Chichester, U.K.

1997.

Schaum Series Vector and Tensor Analysis Mc Graw Hill Company.

K. L. Mir Vector and Tensor Analysis Ilmi Kitab Khana.

Dr. Nawzish Ali Shah Vector and Tensor Analysis


MTH-507 Algebra-I 4(4-0) 80

ALGEBRA –I

Groups and subgroups. Generators and relations. Cyclic group, Cosets and

Lagrange’s theorem. Normalizers and centralizers. Center of a group. Subgroups.

Factor groups, Isomorphism theorems and automorphisms. Commutators.

Permutation groups and Cayley’s theorem, Introduction to Rings, Types of Rings,

Integral domains. Field and its characteristic.

BOOKS RECOMMENDED

I.D. Macdonald, Theory of Groups Oxford University Press

I.N., Herstein, Topics in Algebra, Addison-Wesley.

J.B, Fraleigh, Abstract Algebra. Addison-Wesley.

K.H. Dar, First Step to Abstract Algebra, (2nd

edition 1998). Feroz Sons, 1998.

J.T. Scheik, Linear Algebra with Applications, 1997, McGraw Hill.

A Majeed, Theory of groups, Ilmi Kitab Khana.



Q. Mushtaq, A course in Group Theory, (1993). University Grants Commission, Islamabad.


MTH-509 Point set Topology 4(4-0) 80

POINT SET TOPOLOGY

Set, countable and uncountable sets, partially and totally ordered sets and lattices with

examples, axiom of choice. Topological spaces; subspaces and relative topology, open

sets, closed sets, neighborhood, interior, exterior and limit points, base and sub base,

product spaces, continuous and open mappings, homeomorphism, first and second

axioms of countability, separation axioms, T0, T1, T2,\ T3, T31/2

, T4, spaces, regular and

normal spaces, connectedness various characterizations of connectedness, local

connectedness, components, open covers, compact spaces and their characterization,

continuity, uniform continuity and their relationship with compactness in metric spaces,

limit points sequential compactness. equivalence of different notions of compactness.

BOOKS RECOMMENDED

G.F. Simon, Introduction to Topology and Modern Analysis, 1963, McGraw Hill Book

Company, New York.

J. Willard, General Topology, Addison-Wesley Publishing Company, London.

E. Kreyszig, Introduction to Functional Analysis with Applications, 1978, John Wiley and

Sons.

W. Rudin, Functional Analysis, McGraw Hill Book Company, New York.

N. Dunford and J. Schwartz, Linear Operators (Part-I General Theory), Interscience

Publishers, New York.

A, Majeed, Elements of Topology and Functional Analysis, Ilmi Kitab Khana Lahore


MTH-502 Real Analysis - II 4(4-0) 80

REAL ANALYSIS-II

The Riemann – Stieltjes (R-S) Integrals. Properties of R-S integrals. Functions of

bounded variations. Point wise and uniform convergence of sequences and series of

functions, Weierstrass M-Test, Uniform convergence and continuity. Uniform Convergence

and differentiation, Uniform Convergence and integration. Convergence of improper

integrals. Beta and Gamma functions and their properties. Implicit functions, Jacobians,

Functional dependence. Taylor’s theorem for a function of two variables. Maxmima and

minima of functions of two and three variables. Method of Lagrange Mulitipliers.

Book Recommended Bartle RG Sherbert DR, Introduction to Real Analysis (3

rd edition) 1999, John Wiley New York.

Rudin W. Principles of Mathematics Analysis (3rd

edition) 1986, McGraw-Hill New York.

Brabence RL Introduction to Real Analysis, 1997, PWS Publishing Company.

Gaughan ED. Introduction to Analysis (5th edition) 1997, Books/Cole.

Kaplan W. Advance Calculus (3rd

edition) 1984 Addison-Wesley publishing Company.

Apostol, Mathematical Analysis 6th Prenteng Addison-Wesley Publishing Company.




MTH-504 Algebra-II 4(4-0) 80

ALGEBRA –II

Review of elementary concepts of vector spaces. Linear dependence and

independence of vectors. Vector spaces and subspaces. Quotient spaces. Direct

sum of spaces. Linear transformation. Rank and Nullity of linear transformations.

Algebra of linear transformations and representation of linear transformations as

matrices. Change of bases. Linear functional. Dual spaces and Annihilators.

Eigenvectors, eigenvalues and Cayley–Hamilton theorem. Diagonalization of

Matrices. Inner product spaces. Bilinear, quadratic and Hermit forms.

BOOKS RECOMMENDED

A.M., Tropper, Linear algebra, Thomas Nelson & Sons.

S. Lang, Linear Algebra, Addison-Wesley.

K.R. Hoffman, and Kunze, R., Linear Algebra Prentice –Hall.

I. N., Herstein, Topics in Algebra, Addison- Wesley.

P.R, Halmos, Finite Dimensional Vector Spaces, Von Nostrand.

K.H. Dar, First Step to Abstract Algebra, 2nd

Edition 1998. Feroze Sons Pvt.

J.T. Scheick, Linear Algebra with Applications, 1997. McGraw Hill.

P.M. Chon, Algebra-I and Algebra-II.


MTH-506 Mechanics 4(4-0) 80

MECHANICS

General Motion of a rigid body, Euler’s Theorem and Chasles Theorem. Euler’s

Angles, Moments and Products of Inertia, Inertia Tensor, Euler’s Principal Axes

and Principal Moments of Inertia , Kinetic Energy and Angular Momentum of a

Rigid Body, Momental Ellipsoid and Equimomental Systems, Euler’s dynamical

Equations and their solution in special cases. Heavy asymmetrical Top,

Equilibrium of a Rigid Body, General Conditions of Equilibrium, and Deduction

of Conditions in Special Cases.

BOOKS RECOMMENDED

Synge & Griffith, Principles of Mechanics, McGraw Hill Book Company Inc., New

York.



R.A Becker, Introduction to Theoretical Mechanics, McGraw Hill Book Company,

Inc., New York.


K .L. Mir, Theoretical Mechanics Ilmi Kitab Khana.




MTH-508 Functional Analysis 4(4-0) 80

FUNCTIONAL ANALYSIS

Metric Spaces Definition & examples, Open and closed sets, Convergences, Cauchy

sequence and examples, completeness of a metric space, completeness proofs .Definition and

examples of Normed spaces. Banach spaces. Convergence in Normed space. Basis of

Normed Space. Convex sets. Quotient spaces. Equivalent Norms. Compact Normed space.

Characterization of Banach spaces. Linear operators. Bounded linear operators, Various

characterizations of bounded (continuous) Linear operators. The space of all bounded linear

operators. Linear Functional and their examples. Dual Space and Reflexive space. Inner

product spaces and their examples, The Cauchy-Schwarz inequality. Polarization Identity.

Hilbert spaces, Bessel’s inequality. Gram-Schmidt orthogonalization Process. Minimizing

Vector .Direct Sum of spaces. The Riesz representation theorem. Annihilators and

Orthogonal complements. Direct Decomposition.

RECOMMENDED BOOKS

E. Kreyszig, Introductory Analysis with Applications, John Wilsy, 1978.

J Maddox,.Elements of Functional Analysis, Cambridge ,1970.

G.F. Simmon, Introduction to Topology and Modern Analysis, McGraw-Hill- N. Y,1983.

W. Rudin, Functional Analysis, McGraw-Hill. N.Y.1983.

A. Majeed, Elements of Topology and Functional Analysis Ilmi Kitab Khana Lahore.


MTH-510 Differential Geometry 4(4-0) 80

DIFFERENTIAL GEOMERTRY

The moving trihedron, Arc length; The osculating plane, Curvature and torsion of unit speed

and non unit speed curves, Serret-Frenet formulae. Helices, Spherical indicatericies,

Evolutes & Involutes. Simple surface and coordinate patches. The tangent plane and the

normal planes, the first fundamental form and the metric, coordinate transformations.

Surface curves: the angle between two curves on a surface; Normal curvature Analysis

and geodesic curvature, The second fundamental form, Christoffel symbols. Gauss’s

theorem Egregium. Mean and Gaussian curvatures, Principal curvatures, Euler’s theorem,

Dupin’s indicatericies. Weingarthen Map. Guass-Codazzi equations.

BOOKS RECOMMENDED

R. Millman, and G. Parker. Elements of differential Geometry Prentice Hall Inc.

B., O’ Neill. Elementary Differential Geometry.

D.J. Struik. Lectures on Classical Differential Geometry Addison- Wesley.

A. Goetz, Introduction to Differential Geometry Addison- Wesley.

F. Chorlton. Vector and Tensor Methods Ellis Harwood.




MTH-651 Advanced Group Theory 4(4-0) 80

ADVANCED GROUP THEORY

Introduction to Sets and Structures. Examples of groups. Finite groups. Subgroups.

Permutations and cyclic groups. Isomorphism’s and Homomorphism with separate reference

to Abelian groups. Cosets, Normal groups, Factor groups and Simple groups. Series of group

,s. The Sylow theorems. Groups actions, Free groups and group presentations, Geometric,

Analytic and dynamical applications. A brief introduction to continuous groups and group

representations.

RECOMMENDED BOOKS

J.B. Fraleigh, A First Course in Algebra, Addison-Wesley 1982.

M. Hamermesh, Group Theory, Addisosn-Wesley 1972.

I.N. Herstein, Topics in Algebra, John Wiley 1975.


MTH-653 Advanced Set Theory 4(4-0) 80

ADVANCED SET THEORY

Equivalent sets, Countable and uncountable sets, The concept of cardinal number,

Addition and multiplication of cardinals, Cartesian products as sets of function,

Addition and multiplication of ordinals. Partially ordered sets axiom of choice,

statement of lemma.

BOOKS RECOMMENDED

Frankal, Abstract Set Theory, North-Holland Publishing Company Amsterdam.

Patrick Suppes, Axiomatic Set Theory, Dover Publications, Inc., New York.

P.R. Halmos, Naïve Set Theory, New York, Van Nostrand.

B.Rotman & G.T. Kneebone, The Theory of Sets and Transfinite Numbers, Old bourne

London.

P.R. Halmos, Measure Theory, Von Nostrand, New York.

W. Rudin, Real and Complex Analysis, Mcgraw Hill, New York.

R.G. Bartle, Theory of Integration.

H.L. Royden, Real Analysis.


MTH-655 Mathematical Statistics –I 4(4-0) 80

MATHEMATICAL STATISTICS-I

Interpretations of Probability. Experiments and events, Definition of probability, Finite

sample spaces. Counting methods. The probability of a union of events. Independent events.

Definition of conditional probability. Baye’s’ theorem. Random variables and Discrete



Distributions. Continuous distributions. Probability function and probability density function.

The distribution function, Bivariate distributions, Marginal distributions. Conditional

distributions. Multivariate distributions. Functions of random variables. The expectation of a

random variable. Properties of expectations. Variance, Moments. The mean and the median.

Covariance and correlation. Conditional expectation. The sample mean and associated

inequalities. The multivariate normal distribution.

RECOMMENDED BOOKS

A.M. Mood, Graybill, F.A, Boes, D.C. Introduction to the Theory of Statistics, 3rd

Edition,(McGraw-Hill Book Company New York, 1974).

M. H Degroot,. Probability and Statistics, (2nd

edition), Addison-Wesley Publishing

Company, USA, 1986.

K.V. Mardia, Kent, J.T. Bibby, J.M. Multivariate Analysis. Academic Press New

York,1979.


MTH-607 Continuous Groups 4(4-0) 80

CONTINUOUS GROUPS

Continuous Groups; Gl(n,r),Gl(n,c),So(p,q),Sp(2n); generalities on Continuous Groups;

Groups of isometrics; Introduction to Lie groups with special emphasis on matrix Lie groups;

Relationship of isometrics and Lie group; Theorem of Cartan; Correspondence of continuous

groups with Lie algebras; Classification of groups of low dimensions; Homogeneous spaces

and orbit types; Curvature of invariant metrics on Lie groups and homogeneous spaces.

RECOMMENDED BOOKS

Bredon,G.E, Introduction to compact Transformation groups Academic Press,- 1972).

Eisenhart, L.P. Continuous Groups of Transformations (Priceton U.P.1933).

Pontrjagin ,L. S., Topological groups.(Princeton University Press, 1939).

Hussain Taqdir. Introduction to Topological Groups.(W.B.Saunder’s- Company,1966).

Miller Willard, Jr., Symmetry groups and their application;(Academic Press-

New York and London 1972).


MTH-609 Theory of Modules 4(4-0) 80

THEORY OF MODULES

Definition and examples, Sub modules, Homeomorphisms and quotient modules. Direct

sums of modules. Finitely generated modules, Torsion Modules, Free modules. Basis, Rank

and endomorphism of free modules. Matrices over Rings and their connections with the basis

of free modules. A Module. A Module as the direct sum of a free and a torsion module. Exact



sequences and elementary notions of homological algebra. Noetherian and modules,

Radicals, Semi simple rings and modules.

BOOKS RECOMMENDED

Blyth, T.S., Module theory, Oxford University Press, 1977.

Hartley, B. and Hawkes, T.O. Rings, Modules and Linear Algebra, Chapman and Hall,

1980.

Herstein, I.N. Topics in Algebra, John Wiley and Sons, 1975.

Adamson, J. Rings and Modules.


MTH-611 Algebraic Topology 4(4-0) 80

ALGEBRAIC TOPOLOGY

Path wise connectedness; Notion of homotopy, Homotopy classes. Path homotopy. Path

homotopy classes; Fundamental groups, Covering maps. Covering spaces. Litting properties

of covering spaces, Fundamental group of a circle.

RECOMMENDED BOOKS

Kosniowski, C.A first course in Algebraic Topology (Cambridge University Press)

Greenberg, M.J, Algebraic Topology. A first Course (Benjamin. 1976).

Wallace,A. H. Algebraic Topology. Gemology and Comology.(Benjamin,1968).


MTH-613 Advanced Topology 4(4-00 80

ADVANCED TOPOLOGY

Compactness in metric spaces, Limit point, Compactness, Sequential compactness and their

various characterizations, Equivalence of different notions of compactness. Connectedness,

various characterizations of connectedness, Connectedness and T()spaces, Local

connectedness, Path-connectedness, Components. Homotopic maps, Homotopic paths, Loop

spaces, Fundamental groups, Covering spaces, the lifting theorem, Fundamental groups of

the circle () etc.Chain complex, Notion of homology.

RECOMMENDED BOOKS

Greenberg, M.J. Algebraic Topology, A first Course, The Benjamin/Commings

Publishing Company,1967.

Wallace, A.H. Algebraic Topology, Homology and eohomology. W.A. Benjamin, Inc

New York, 1968.

Gemignani, M.C., Elementary Topology, Addison-Wesley Publishing Company, 1972.




MTH-615 Numerical Analysis-I 4(4-0) 80

NUMERICAL ANALYSIS- I

Introduction, Computation error and error analysis, Study of various Iterative methods to

solve non-linear equations with analysis of error, Convergence and stability of Bisection,

False position , Secant, Newton- Rephson and Fixed point methods, Acceleration of

convergence by Aitken method, Solution of system of linear equations by LU

decomposition method, Cases of failure, Iterative methods, (Jacobi, Gauss Seidel, SOR,

SUR) and their convergence analysis , Ill conditioned systems and Condition number,

Interpolation: Review of simple interpolation for equally spaced data, Interpolation by

Gauss forward/ backward methods, Bessel and Stirling method with error analysis,

Lagrange Interpolation and Newton divided differences formula with error analysis,

Interpolation by Spline functions (up to Cubic spline), methods of Least squares, Numerical

differentiation, Numerical integration for equally spaced data (Newton cotes formula and its

special cases e.g. Trapezoidal Rule and Simpson’s rules) and for unequally spaced

data(using Lagrange and divided differences formula of interpolation), Gaussian quadrature

using a system of orthogonal polynomials (Legendre and Laguere polynomials).

BOOKS RECOMMENDED

C. Gerald, Applied Numerical Analysis, Addison-Wesley Publishing Company, 1978.

A. Balfour & W.T. Beveridge, Basic Numerical Analysis with Fortran, Heinemann

Educational Books Ltd. (1977).

Shan and Kuo, Computer Applications of Numerical Methods (Addison – Wesley) National

Book Foundation, (1972), Islamabad.

R. L. Burden and J. D. Faires, Brooks /Cole Publishing Company, New York.


MTH-617 Linear Algebra 4(4-0) 80

LINEAR ALGEBRA

Review of elementary concepts of Vector spaces. Linear dependence and Independence of

vectors. Vector spaces and subspaces. Quotient spaces. Direct sum of spaces. Linear

transformation. Rank and Nullity of linear transformations. Algebra of linear transformations

and representation of linear transformations as matrices. Change of bases. Linear functional.

Dual spaces and Annihilators. Eigenvectors, Eigenvalues and Cayley–Hamilton theorem.

Digitalization of matrices. Inner product spaces. Bilinear, Quadratic and Hermit Ian forms.

BOOKS RECOMMENDED K.H. Dar, First Step to Abstract Algebra, (2

nd Edition 1998). Feroze Sons Pvt.

J.T. Scheick, Linear Algebra with Applications, 1997. McGraw Hill.

K.R. Hoffman, and Kunze, R., Linear Algebra Prentice –Hall.

A.M., Tropper, Linear algebra, Thomas Nelson & Sons.

S. Lang, Linear Algebra, Addison-Wesley.

I. N., Herstein, Topics in Algebra, Addison- Wesley.

P.R, Halmos, Finite Dimensional Vector Spaces, Von Nostrand.

P.M. Chon, Algebra-I and Algebra-II.




MTH-619 Rings & Fields 4(4-0) 80

RINGS AND FIELDS

Definitions and basic concepts, homeomorphisms, homomorphism theorems. Polynomial

rings. Unique factorization domain, factorization theory. Euclidean domain, arithmetic in

Euclidean domains, Extension fields, Algebraic and transcendental elements, simple

extension, Introduction to Galois theory.

RECOMMENDED BOOKS


1980.





MTH-621 Fluid Mechanics- I 4(4-0) 80

FLUID MECHANICS -I

Real fluids and ideal fluids, Velocity of a fluid at a point, Streamlines and path lines, Steady

ad unsteady flows, Velocity potential, Vorticity vector, Local and particle rates of change,

Equation of continuity. Acceleration of a fluid, Conditions at a rigid boundary, General

Analysis of fluid motion Euler’s equations of motion, Bernoulli’s equations steady motion

under conservative body forces, Some potential theorems, impulsive motion. Sources, Sinks

and doublets, Images in rigid infinite plane and solid spheres, Axi-symmetric flows,

Stokes’s stream function.

Stream function, Complex potential for two-dimensional, Irrational, Incompressible flow,

Complex velocity spotential for uniform stream. Line sources and line sinks, Line doublets

image systems, Miline-Thomson circle theorem, Blasius’s Theorem.

BOOKS RECOMMENDED

Chorlton, F., Text Book of fluid Dynamics D. Van No strand Co. Ltd.1967.

Thomson, M., Theoretical Hydrodynamics, Macmillan Press, 1979.

Jaunzemics, W., Continuum Mechanic, Macmillan Company, 1967.

Landau, L.D, and Lifshitz, E.M., Fluid Mechanics, Pergamon Press, 1966.

Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge University Press, 1969.

Brvce R. Munson and Donald F. Young, Fundamentals of Fluid Mechanics, Department of

Engineering Science and Mechanics, Department of Mechanics Engineering Lowa State

university Amos Lowa USA.




MTH-623 Advanced Mathematical Methods 4(4-0) 80

Advanced Mathematical Methods

Theory of Sturm-Liouville Problems. Linear homogeneous differential equations of order n.

Fundamental set of solutions. Linearly dependent and independent solutions. Wronskian

Determinant. Adjoint and self-adjoint Equations. Self-adjoint operator. Symmetric operator.

Lagrange’s identity. Green’s identity. Eigenvalue problem. Eigenfunctions and eigenvalues.

Self-adjoint eigenvalue problem..

Orthogonality of eigenfunctions. Real eigenvalues. Regular, Periodic and singular Sturm-

liouvill systems. Orthogonal Sets of functions. Expansion of functions in terms of

eigenfunctions.

Power Series, Solutions of Legendre’s Equation, Legendre’s polynomials Generating

function; Rodrigue’s formula, Recurrsion relations. Orthogonality and normality of

Legendre’s Polynomials, Legendre’s Series. Bessel’s equation, Bessel’s Functions,

Generating function, Recurring relations, Orthogonality of Bassel’s function, Bessel’s series

Green’s function methods applied to ODEs. Green’s function in one and two dimensions.

Integral Equations. Formulation and classification of integral equations. Degenerate Kernels.

Methods of successive approximations.

BOOKS RECOMMENDED

I. Stakgold, Boundary Value Problems of Mathematical Physics, Vol. I, II. Macmillan.

Lal Din Baig Methods of Mathematical Physics 2000.

H.Sagan, Boundary and Eigenvalue problems in Mathematical Physics.

E.L Butkov, Mathematical Physics, Addison-Wesley

G. Arfken, Mathematical Methods for Physics, Academic Press.

R.P kanwal, Linear Integral Equations.

My-Tung & Debnath, Partail Differential Equations

MTH-625 Special Theory of Relativity 4(4-0) 80

SPECIAL THEORY OF RELATIVITY

Historical background and fundamental concepts of Special Theory of Relativity. Lorentz

transformations (for motion along axis). Length contraction. Time dilation and simultaneity.

Velocity addition formulae. 3-dimensional Lorentz transformations. Introduction to 4-vector

formalism. Lorentz transformations in the 4 vector formalism. The Lorentz and Poincare

groups. Introduction to classical Mechanics. Minkowski spacetime and null cone. 4-

velocity, 4 acceleration 4-momentum and 4-force. Application of Special Relativity to

Doppler shift and Compton Effect. Particle scattering. Binding energy, Particle production

and decay. Electromagnetism in Relativity. Electric current. Maxwell’s equations and

electromagnetic waves. The 4-vector formulation of Maxwell’s equations. Special Relativity

with small acceleration.

BOOKS RECOMMENDED A. Qadir, Relativity: An Introduction to the Special Theory, World Scientific, 1989.

R. D’ Inverno, Isntroduction Einstein’s Relativity, Oxford University Press. 1992.

H. Goldstein, Classical Mechanics, Addison Wesley, New York, 1962.

J.D. Jackson, Classical Relativity, Springer-Verlag, 1977.

J.G. Taylor, Special Theory of Relativity




MTH-627 Operations Research 4(4-0) 80

OPERATIONS RESEARCH

LINEAR PROGRAMMING

Mathematical modeling. Formulation and graphical solution. Analytical solution. Simplex

method. Two- phase and M-technique for Linear programs.

Duality. Duality simplex method. Sensitivity Analysis.

TRANSPORTATION PROBLEMS

Definition. Various methods including North –West Corner method. Least –cost method and

Vogel’s approximation. The Assignment model. Application to Networks. Shortest- Route

Algorithm for acyclic and cyclic networks. Maximal-flow problems.

INTEGER PROGRAMMING

Definition and formulation- Cutting-Plane Algorithm and Branch-and Bound

method,Application. The mixed Algorithm, Zero-one polynomial programming.

BOOKS RECOMMENDED

Hamdy A Taha An Introduction to Operations Research Macmillan Publishing Company

Inc, New York, 1987.

S Kalavathy Operations Research Vikas Publishing House (P) Ltd.

F. S Hiller and G. J Liebraman, Operational Research CBS Publisher and distributors, New

Delhi, 1974.

C. M Harvey, Operation Research, North Holland, New Delhi

Prof. Sr. Saeed Akhtar Bhatti Operations Research: An Introduction

Krajewsky and Ritzman Operations Management Strategy and Analysis.


MTH-629 Quantum Mechanics 4(4-0) 80

QUANTUM MECHANICS

Basic postulates of Quantum mechanics. State vector. Formal properties of quantum

mechanical operators. Eigenvalues and Eigenstates, Simple harmonics oscillator,

Schrodinger representation. Heisenberg equation of motion Schrodinger equation. Potential

step, Potential hydrogen atom. Matrix representation of angular momentum an spin. Time

independent perturbation theory, Degeneracy. The Stark effect. Introduction to relativistic

Quantum Mechanics.

RECOMMENDED BOOKS

Fayyazuddin and Riazuddin, Quantum Mechanics (World Scientific 1990).

Merzbache, E. Quantum Mechanics, John Wiley (2nd

edition) 1970.

Liboff, R.L Introductory Quantum Mechanics, Oxford University Press.




MTH-631 Soft Ware Engineering 4(4-0) 80

SOFT WARE ENGINEERING

Introduction to Software. Designing phases of Software, Different Phase used in Software

Development Process. Applying UML (Unified Modeling Language) & Patterns. Software

Types, Different kinds of Development Cycles. Iterative , Water Fall, Big Bang Techniques.

Software Testing, Black Box Testing, White Box Testing. Concepts of Object Oriented

Programming. Difference Between Structured Programming & Object Oriented

Programming. Advancements of OOPS.

RECOMMENDED BOOKS

Applying UML & Patterns By Grasp

Practitioner’s approach towards Software Engineering by Pressman

Applying UML & Patterns by Craig Larman


MTH-606 Rings & Modules 4(4-0) 80

RINGS & MODULES

Definition and examples, Submodules, Homomorphisms and Quotient modules. Direct sums

of modules. Finitely generated modules, Torsion Modules, Free modules. Basis, rank and

endomorphism of free modules. Matrices over Rings and their connections with the basis of

free modules. A Module. A Module as the direct sum of a free and a torsion module. Exact

sequences and elementary notions of homological algebra. Noetherian and modules,

Radicals, Semi simple rings and modules.

BOOKS RECOMMENDED



1980.

R. B. J. E. Allenly, Rings Fields and Groups, An Introduction to Abstract Algebra,

Edward Arnold, 1985.


Lal Din Baig Methods of Mathematical Physics 2000.





MTH-608 Theory of Numbers 4(4-0) 80

THEORY OF NUMBERS

Algebraic Numbers and integers, Units and Primes in R(v). Ideals. Arithmetic of Ideals congruences. The norm of a Ideal. Prime Ideals. Units of Algebraic number field.

APPLICATION TO RATIONAL NUMBER THEORY

Equivalence and class number. Cyclotomic field K Fermat’s equation. Kummer’s theorem, the q equation X2 + 2=Y3, pure cubic fields. Distribution of Primes and Riemann Zets function, the prime number theorem.

BOOKS RECOMMENDED

W.J. Leveque, Topics in Number Theory, Vol. I and II Addison-Wesley Publishing Co, 1956.

Shailesh Shirali, C. S Yogananda, Number theory Universities press.

Steven Miller Ramin Takloo-Bighash, An Introduction to modern Number Theory Publishing Princeton

Neville Robbins, Beginning Number Theory (2nd edition), Jones and Bartlett.


MTH-610 Mathematical Statistics- II 4(4-0) 80

MATHEMATICAL STATISTICS- II

Statistical inference. Maximum likelihood estimators. Properties of maximum likelihood

estimators. Sufficient statistics. Jointly sufficient statistics. Minimal sufficient sufficient

statistics. The sampling distribution of a statistic. The Chi square distribution. Joint

distribution of the sample mean and sample variance. The t distribution. Confidence

intervals. Unbiased estimators. Fisher information. Testing simple hypotheses. Uniformly

most powerful tests. The t test. The F distribution. Comparing the means of two normal

distributions. Tests of goodness of fit. Contingency tables. Equivalence of confidence sets

and tests. Kolmogorov- Smirnov tests. The Wilcoxon Signed-ranks tests. The Wilcoxon-

Mann-Whitney Ranks test.



RECOMMENDED BOOKS

Mood, A.M. Graybill, F.A . Boes, D.C. Introduction to the Theory of Statistics, (2nd

edition), McGraw-Hill Book Company New York ,1986.

Degroot, M.H. Probability and Statistics, (2nd

edition) Addison Wesley Company New

York 1986.

Walpole-Myers. Myers. Ye Probability and Statistics (7th

edition)

K. V. Mardia, Kent, J. T. Bibby, J. M. Multivariate Analysis Academic Press New York

1979.

Allen. T Craig, Robert V. Hogg, Introduction to Mathematical Statistics 5th

edition

publish by Pearson education Singapore (Pvt) Ltd


MTH-612 Numerical Analysis-II 4(4-0) 80

NUMERICAL ANALYSIS-II

Methods of least squares, Numerical Integration for equally spaced data, Newton

cotes formula and its special cases e.g. Trapezoidal Rule Simson’s Rules, Gaussian

quadrature using a system of orthogonal, Polynomials (Legender and Laguere Ploynomials,

Numerical Differentiation, Difference Equations, Differential Equations, Euler’s Method,

Improved Euler’s Methods. Mid point Formula, Heun’s Method,

BOOKS RECOMMENDED

Johnson L., and Dean, R.; Numerical Analysis, Addison Wesley.

James, M.L., Smith, G.M. & Woford, J.C., Applied Numerical Methods for Digital

Computation, Harper and Row, Publications.

Ralston, A & Philips, R.A. First Course in Numerical Analysis, McGraw Hill.

Froeberg , C.E. Introduction to Numerical Analysis, Addison Wesley.

Scarborough , J.B., Numerical Mathematical Analysis , John Hopkins Press.

M. Iqbal, Numerical Analysis , National Book Foundation.

J.H. Wilkinson, Eigenvalue Problems, Oxford University Press.

Aitkinson , Elementary Numerical Analysis.


MTH-614 Special Functions 4(4-0) 80

SPECIAL FUNCTIONS

Sturm-Liouville theory of DEs, Basic properties of a SL System, Orthogonality, Reality and

uniqueness, Sturm’s comparison theorem, completeness of Eigenfunctions Via Rayleigh

quotient, Bessel functions, Legendre Polynomial and their Generating functions and

properties, Hermite equation and functions and their properties, Laguerre Equation and

functions and their properties, chibeicif function, Hypergeomtric Differential Equations and

Functions, Gamma and Beta Functions



RECOMMENDED BOOKS

G.B. Arfken And H. J. Weber, Mathematical Methods for Physicists, CUP New York.

B. G. Korenev, Bessel Functions and their Applications

R. E. Attar, Special Functions and Orthogonal Polynomials

G. N. Watson, A Treatise on the Theory of Bessel Functions


MTH-616 Theory of Optimization 4(4-0) 80

THEORY OF OPTIMIZATION

Introduction to optimization. Relative and absolute extreme. Convex. Concave and unimodal

functions. Constants. Mathematical programming problems. Optimization of one, two and

several variables functions and necessary and sufficient conditions for their optima.

OPTIMIZATION BY EQUALITY CONSTRAINTS

Direct substitution method and Lagrange multiplier method, necessary and sufficient

conditions for an equality-constrained optimum with bounded independent variables.

Inequality constraints and Lagrange multipliers. Kuhn-Tucker Theorem. Multidimensional

optimization by Gradient method. Convex and concave programming, Calculus of variation

and Euler Language equations, Functions depending on several independent variables.

Variational problems in parametric form. Generalized mathematical formulation of dynamics

programming. Non-Linear continuous models, Dynamics programming and Variational

calculus. Control theory.

RECOMMENDED BOOKS

Gotfried B.S and Weisman, J. Introduction to Optimization Theory (Prentice-Inc. New

Jersey,1973).

Elsgolts. L. Differential Equations and the Calculus of Variations (Mir Publishers-

Moscow,1970).

Wismer D.A and Chattergy R. Introduction to Nonlinear Optimization (North - Holland,

New York,1978).

Intriligator M.D. Mathematical Optimization and Economic Theory(Prentice-Hall, Inc,

New Jersey,1971).


MTH-619-P Project 4(4-0) 80


% of the class strength and only to those who obtain at least 65 % marks on the basis of their





MTH-618 Fluid Mechanics -II 4(4-0) 80

FLUID MECHANICS -II

Vortex motion, Line Vortex, Vortex row Image System, Kelvin’s minimum energy

theorem, Uniqueness theorem, Fluid streaming past a circular cylinder, Irrational

motion produced by a vortex filament. The Helmholtz vorticity equation,

Karman’s vortex-street.

Constitutive equations; Navier- Stoke’s equations; Exact solution of Navier- Stoke’s

equations; Steady unidirectional flow; Poiseuille flow; Couette flow; Unsteady undirectional

flow, Sudden motion of a plane boundary in a fluid at rest; Flow due to an oscillatory

boundary; Equations of motion relative to a rotating system; Ekman flow; Dynamical

similarity of turbulent motion.

BOOKS RECOMMENDED

L.D. Landan & E. M. Lifshitz, Fluid Mechanics, Pergamon Press, 1966.

Batchelor, G.K. An Introduction to Fluid Dynamics, Cambridge University Press, 1969.

Walter Jaunzemis, Continuum Mechanics, McMillan Company, 1967.

Milne-Thomas, Theoretical Hydrodynamics, McMillan Company, 1967.

D. J Tritton, Physical Fluid Dynamics 2nd

edition Oxford


MTH-620 Partial Differential Equations 4(4-0) 80

PARTIAL DIFFERENTIAL EQUATIONS

Basic Concepts and Definitions, Formation and Classification of partial differential equations

(PDEs). Partial differential equations of the first order. Nonlinear PDEs of first order.

Applications of first order PDEs. Partial differential equations of second order: Mathematical

formation of heat, Laplace and wave equations. Classification of second order PDEs.

Boundary and initial conditions. Characteristics. Method of Characteristics. Reduction to

various Canonical (Normal) forms. And the general solutions of PDEs. Methods of

separation of variables (Product Method ) for solving PDEs like elliptic, parabolic and

hyperbolic equations. The Cauchy Problem. Cauchy’s Problem for hyperbolic system in two

independent variables with application to wave equations. Laplace, Fourier and Hankel

transform for the solution of PDEs and their application to boundary value problems.

RECOMMENDED BOOKS

E.I. Butkov, Mathematical Physics, Addison-Wesley.

H. Sagan, Boundary and Eigenvalue Problems in Mathematical Physics.

My-Tung & Debnath, Partial Differential Equations.

G. Arken, Mathematical Methods for Physics, Academic Press.

I. Stakgold, Boundary Value Problems of Mathematical Physics, Vol. I, II Macmillan.



Sneddon, I.N., Elements of Partial Differential Equations, McGraw-Hill Book Company,

1987.

Dennemyer, R., Introduction to Partial Differential Equations and Boundary Value Problems,

MeGraw-Hill Book Company,1968.

Himi, M And Millerl, W.B., Boundary Value Problems and Partial Differential equations

PWS-Kent Publishing Company, Boston, 1992.

Chester, C. R., Techniques in Partial Differential equations MeGraw-Hill Book Company,

1971.

Haberman, R., Elementary Applied Partial Differential Equations, Prentice Hall, Inc. New

Jersey, 1983.

Zauderer E., Partial differential Equations of Applied Mathematics, John Wiley & Sons,

Englewood Cliff, New York, 1983

J. D. Logan, Partial Differential Equations, Second Edition, Springer-Verlag, 2004


MTH-622 Theory of Elasticity 4(4-0) 80

THEORY OF ELASTICITY

Cartesian tensors, Analysis of stress and strain, Generalized Hooke’s law; crystalline

structure, Point groups of crystals, Reduction in the number of elastic moduli due to crystal

symmetry; Equations of equilibrium; Boundary conditions, ompatibility equations; Plane

stress and plane strain problems; Two dimensional problems in rectangular and polar co-

ordinates; torsion of rods and beams.

RECOMMENDED BOOKS

S. P. Timoshenko And J.N. Goodier, Theory of Elasticity, 3rd

edition, McGraw Hill Book

Company, 1970, 1987.

A. P. Boresi And K. P. Chong, Elasticity iri Engineering Mechanics, 2nd

edition, John

Wiley & Sons, 2000.

A.C. Ugural And S.K. Fenster, Advanced Strength and Applied Elasticity, 2nd

edition

Elsevier Science Publishing Co., Inc., 1987.

Adel S. Saada, Elasticity: Theory and Applications, Second Edition, Krieger Publishing,

Malabar, Florida, 1993.

Abdel-Rahman Ragab & Salah Eldin Bayoumi, Engineering Solid Mechanics: Fundamentals

and Applications, CRC Press, Boca Raton, Florida, 1999.




MTH-624 Electromagnetism 4(4-0) 80

ELECTROMEGNETISM

Electrostatics and the solution of electrostatics problems in vacuum and in media,

Electrostatic energy, Electro currents, The magnetic field of steady currents. Magnetic

properties of matter. Magnetic energy, Electromagnetic Introduction, Maxwell’s equations,

Boundary Value Potential Problems in two dimensions, Electromagnetic Waves, Radiation,

Motion of electric charges.

RECOMMENDED BOOKS

Reitz, J. R. & Milford, F. J., Foundation of Electromagnetic Theory, Addison-Wesley

1969.

Panofsky, K.H. and Philips, M., Classical Electricity and Magnetism, Addison Wesley,

1962.

Corson, D. and Lerrain, P., Introduction to Electromagnetic Fields and Waves, Freeman,

1962.

Jackson, D.W., Classical Electrodynamics, John-Wiley.

Ferraro, V.C.A., Electromagnetic Theory, the Athlone Press, 1968.


MTH-615 Mathematical Statistics- II 4(4-0) 80

MATHEMATICAL STATISTICS- II

Statistical inference. Maximum likelihood estimators. Properties of maximum likelihood

estimators. Sufficient statistics. Jointly sufficient statistics. Minimal sufficient sufficient

statistics. The sampling distribution of a statistic. The chi square distribution. Joint

distribution of the sample mean and sample variance. The t distribution. Confidence

intervals. Unbiased estimators. Fisher information. Testing simple hypotheses. Uniformly

most powerful tests. The t test. The F distribution. Comparing the means of two normal

distributions. Tests of goodness of fit. Contingency tables. Equivalence of confidence sets

and tests. Kolmogorov- Smirnov tests. The Wilcoxon Signed-ranks tests. The Wilcoxon-

Mann-Whitney Ranks test.

RECOMMENDED BOOKS

Mood, A.M. Graybill, F.A . Boes, D.C. Introduction to the Theory of Statistics, (2nd

edition), McGraw-Hill Book Company New York ,1986.

Degroot, M.H. Probability and Statistics, (2nd

edition) Addison Wesley Company New

York 1986.

Walpole-Myers. Myers. Ye Probability and Statistics (7th

edition)

K. V. Mardia, Kent, J. T. Bibby, J. M. Multivariate Analysis Academic Press New York

1979.



Allen. T Craig, Robert V. Hogg, Introduction to Mathematical Statistics 5th

edition

publish by Pearson education Singapore (Pvt) Ltd


MTH-616 Numerical Analysis-II 4(4-0) 80

NUMERICAL ANALYSIS-II

Methods of least squares, Numerical Integration for equally spaced data, Newton

cotes formula and its special cases e.g. Trapezoidal Rule Simson’s Rules, Gaussian

quadrature using a system of orthogonal, polynomials (Legender and Laguere Ploynomials,

Numerical Differentiation, Difference Equations, Differential Equations, Euler’s Method,

Improved Euler’s Methods. Mid point Formula, Heun’s Method,

BOOKS RECOMMENDED

Johnson L., and Dean, R.; Numerical Analysis, Addison Wesley.

James, M.L., Smith, G.M. & Woford, J.C., Applied Numerical Methods for Digital

Computation, Harper and Row, Publications.

Ralston, A & Philips, R.A. First Course in Numerical Analysis, McGraw Hill.

Froeberg , C.E. Introduction to Numerical Analysis, Addison Wesley.

Scarborough , J.B., Numerical Mathematical Analysis , John Hopkins Press.

M. Iqbal, Numerical Analysis , National Book Foundation.

J.H. Wilkinson, Eigenvalue Problems, Oxford University Press.

Aitkinson , Elementary Numerical Analysis.


MTH-617 Special Functions 4(4-0) 80

SPECIAL FUNCTIONS

Sturm-Liouville theory of DEs, Basic properties of a SL System, Orthogonality, Reality and

uniqueness, Sturm’s comparison theorem, completeness of Eigenfunctions Via Rayleigh

quotient, Bessel functions, Legendre Polynomial and their Generating functions and

properties, Hermite equation and functions and their properties, Laguerre Equation and

functions and their properties, chibeicif function, Hypergeomtric Differential Equations and

Functions, Gamma and Beta Functions

RECOMMENDED BOOKS

G.B. Arfken And H. J. Weber, Mathematical Methods for Physicists, CUP New York.

B. G. Korenev, Bessel Functions and their Applications

R. E. Attar, Special Functions and Orthogonal Polynomials

G. N. Watson, A Treatise on the Theory of Bessel Functions




MTH-618 Theory of Optimization 4(4-0) 80

OPTIMIZATION THEORY

Introduction to optimization. Relative and absolute extreme. Convex. Concave and unimodal

functions. Constants. Mathematical programming problems. Optimization of one, two and

several variables functions and necessary and sufficient conditions for their optima.

OPTIMIZATION BY EQUALITY CONSTRAINTS

Direct substitution method and Lagrange multiplier method, necessary and sufficient

conditions for an equality-constrained optimum with bounded independent variables.

Inequality constraints and Lagrange multipliers. Kuhn-Tucker Theorem. Multidimensional

optimization by Gradient method. Convex and concave programming, Calculus of variation

and Eulre Language equations, Functions depending on several independent variables.

Variational problems in parametric form. Generalized mathematical formulation of dynamics

programming. Non-Linear continuous models Dynamics programming and Variational

calculus. Control theory.

RECOMMENDED BOOKS

Gotfried B.S and Weisman, J. Introduction to Optimization Theory (Prentice-Inc. New

Jersey,1973).

Elsgolts. L. Differential Equations and the Calculus of Variations (Mir Publishers-

Moscow,1970).

Wismer D.A and Chattergy R. Introduction to Nonlinear Optimization (North -

Holland, New York,1978).

Intriligator M.D. Mathematical Optimization and Economic Theory(Prentice-Hall, Inc,

New Jersey,1971).


MTH-628 Project 4(4-0) 80


% of the class strength and only to those who obtain at least 65 % marks on the basis of their





MTH-626 C++ 4(4-0) 80

PROGRAMMING LANGUAGE C++.

Object Oriented Programming using C++. Declaring Variable, Designing Functions,

Designing Classes, Using Built in Functions and Libraries.

Introduction I:- History of C++, writing C++ Program, structure, preprocessor, Header file,

Main function, Increment operators++, data types, Declaration of the variable, Initialization

of the variable, Arithmetic operators, arithmetic Expression, order of precedence of

operation.

Introduction 2:- Basis input / output, cout<< object, the escape sequence, the end line, setw

manipulator, Assignment operator, the cin>> operator. Compound assignment, increment and

decrement operator, the comment statement, the conditional statement, loops statement,

arrays, structures, functions part I and part II, pointers, inheritance, and polymorphism part I

and II, Files graphics, bit wise operators.

RECOMMENDED BOOKS

Object Oriented Programming Using C++ by Robert Lafore 3rd

Edition.

Aikman Series. C.M Aslam T A Qurashi. Urdu bazaar Lahore.


MTH-508 Advanced Functional Analysis 4(4-0) 80

ADVANCED FUNCTIONAL ANALYSIS

Hahn-Banach theorem, adjoint operator, uniform boundednes theorem, strong and weak

convergence, convergence of sequences of operators and functionals, open mapping theorem,

closed graph theorem, Banach fixed point theorem and its applications, spectral theory in

fininte dimensional normed spaces, spectral perperties of bounded linear operators, compact

linear operators and their properties, spectral properties of compact linear operators on

normed spaces.

RECOMMENDED BOOKS

E. Kreyszig, Introductory Analysis with Applications, John Wilsy, 1978.

J Maddox,.Elements of Functional Analysis, Cambridge ,1970.

W. Rudin, Functional Analysis, McGraw-Hill. N.Y.1983.

N. L. Carothers, Banach Space Theory, Cambridge University Press, 2004.

The End

Documents

Scheme of Studies - Rising Education · MTH-506 Mechanics 4(4-0) MTH-508 Functional Analysis 4(4-0) MTH-510 Differential Geometry 4(4-0) Semester 7 PURE MTHEMATICS Course Code Course